Math 116 Chapter Notes
Section 10.1-10.4
The Mathematics of Population Growth:
1. Population Growth: Has a very broad meaning, due primarily to the way that both
words-population and growth are interpreted nowadays. Population nowadays can mean
human, to any collection of objects (like penguins and dollars in a bank). Growth on the
other hand can be positive or negative. If growth is negative, it is decay
2. Two types of Population Growth:
A. Continuous Growth- the dynamics of change are in effect all the time-every hour,
every minute, every second, there is change.
B. Discrete Growth- is the most common and natural way by which a population changes.
3. Transition- when there is a sudden change in the population growth.
4. Population Sequence: The flow of a particular population over time is a sequence of
, ,
,...
,
numbers P0 P1 P3 Pn 1 Pn .
5. Two types of "Time-Series" graphs will be used to describe the population sequence.
The horizontal axis usually represents time and the vertical axis usually represents the
size of the population. Page 341.
A. Scatter Plot- a set of dots that represents a point (x,y) where x = time, and y = size of
population.
B. Line Graph- is a scatter-plot where all the dots are connected with straight lines.
The Linear Growth Model: Is the simplest of all models of population growth. In this
model, in each generation the population increases (or decreases) by a fixed amount
called a common difference. Page 343 see example.
6. Arithmetic Sequence: is just a numerical description of a population growing
according to a linear growth model-informally.
7. Linear Growth Model:
A. P N  P N 1  d (recursive formula)
B. PN  P0  Nxd (explicit formula)
8. Arithmetic Sum Formula:
 n1 
  Ai  N
 i 0 
A0  A1  A2  ...  An1  2
For example;
S  5  12  19  ...  6998
S  6998  6991  6984  ...  5
S
1000 7003
2
So simply, to find the sum of terms of an arithmetic sequence, add the first term and the
last term, multiply the result by the number of terms, and then divide by two. This rule
only applies when you are adding consecutive terms of an arithmetic sequence.
Example;
Suppose we need to compute the arithmetic sum 4  13  22  31  40  ...  922 .
Here the first and last terms are given-the missing piece of the puzzle is the number of
terms in this sum. To calculate this use the common difference of d = 9 and the explicit
formula.
922  4  9n  1 , solve for n. This implies that n = 103.
S
926 103
2
8. The Exponential Growth Model: Is the most important and widely used (some would
say overused) model of population growth. Exponential growth is based on the idea of a
constant growth rate-in each transition the population changes by a fixed factor called the
common ratio.
Note: The difference between linear and exponential growth-in linear growth we add a
fixed constant and in exponential growth we multiply by a fixed constant.
Geometric Sequence: Is a sequence defined by repeated multiplication-every term in the
sequence after the first is obtained by multiplying the preceding term by a fixed amount
"r". This constant factor "r" is called the common ratio of the geometric sequence-it is
the ratio of two consecutive terms in the sequence.
Exponential Growth Formulas:
P
n
 r  Pn 1
P  P r
n
n
0
When r >1, the terms of the sequence get bigger and we have real (positive) growth, but
when 0 < r < 1, the terms of the sequence get smaller and we have exponential decay.
Example of exponential growth can be with money....
P  P 1i 
n
n
0
Where P0 is the principal, and i annual interest rate, n is the number of years the money is
left in the account.
General compounding formula, when you want to compound either daily, monthly,
semiannually, quarterly.
nt
 r
A  P 1 
 n
0
. Where r/n is the periodic interest rate, n is the compounding period,
r is the annual rate, p is the principal, t is time in years.
The Geometric Sum Formula:
a  ar  a r  ...  a r
2
n 1



a r 1
n
r 1
9. The Logistic Growth Model: When dealing with animal populations, the two models
listed above are inadequate. Linear models growth between transitions is constant, so
this works on (garbage, consumer goods, sales figures, and so on). Exponential models
represent cases of money left in a bank account, and maybe some early stages of animal
populations.
With logistic growth, the rate of growth often depends on the density of the population.
Small populations have plenty of room to spread out thus there growth rates are high,
however, as the population density increases there is less room to grow and there is more
competition for resources-so the growth rate tends to taper off. So here the populations
growth rate is negatively impacted by the population's density. This principal is
particularly important in cases where the population is confined to a limited environment.
This is referred to as a habitat.
Logistic Growth Equation:
P
n 1
 r  1  Pn  Pn
Usually used to describe the growth of Biological populations whose growth rate is in
direct proportion to the amount of space available in the population's habitat. When
confined to a single-species habitat, many animal populations (including human) are
governed by the logistical model or simple variations of it.